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The position vector of a particle $\vec R$ as a function of time is given by $\vec R = 4\sin(2\pi t)\,\hat i + 4\cos(2\pi t)\,\hat j$, where R is in meters, t is in seconds and $\hat i$ and $\hat j$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of particle?
A
Path of the particle is a circle of radius 4 meter
B
Acceleration vector is along $-\vec R$
C
Magnitude of acceleration vector is $\frac{v^2}{R}$ where v is the velocity of particle
D
Magnitude of the velocity of particle is 8 meter/second
Detailed Solution
$x^2 + y^2 = 16$, so the path is a circle of radius 4 m, and the acceleration is centripetal (along $-\vec R$) with magnitude $\frac{v^2}{R}$.
$\vec v = \frac{d\vec R}{dt} = 8\pi\cos 2\pi t\,\hat i - 8\pi\sin 2\pi t\,\hat j$
$|\vec v| = 8\pi$ m/s, not 8 m/s — so that statement is wrong.
$\vec v = \frac{d\vec R}{dt} = 8\pi\cos 2\pi t\,\hat i - 8\pi\sin 2\pi t\,\hat j$
$|\vec v| = 8\pi$ m/s, not 8 m/s — so that statement is wrong.